How To Really Solve This Tricky Algebra Problem? (X) - (x^x)⁴ = 64; x = ??; where x is a real number.

Welcome to the tenth entry in the tricky algebra problem series. This series celebrates the joy of algebra, where the featured problems range from beginner to advanced levels. This time, we are dealing with a relatively simple exponential equation of the following form:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: x^x^4= 64 where x ∈ R
Problem statement — Math illustrated by the author

Given this equation, your task is to figure out the value of ‘x’. Please note that ‘x’ is a real number. Do you think you can solve this problem?

Spoiler Alert

If you wish to solve this problem on your own, I suggest that you pause reading this essay at this point and do so now. The reason for this is that I will be explicitly discussing the solution to this problem beyond this section.

Once you are done with your attempt, you may return to this essay and continue reading to compare our approaches. All the best.

This essay is supported by Generatebg

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*Important* Update Post Publishing — Author’s Note

The below procedure (although appearing to be correct on the surface) is not per convention. As readers Federico Kereki and William Vaughn pointed out, exponential towers like these are solved top-down.

In the below procedure, I have solved the problem bottom-up (which is not per convention). To correct this, I have now updated the essay with a section titled “Corrected Solution”.

If you are interested in reading the correct solution directly, please scroll down to the above-mentioned section. If you are interested in reading the (incorrect) solution below, please feel free as well.

I am letting the initial (incorrect) solution stand to keep a record of my error, and also as a marker to show how tricky this problem can get. Consequently, I am also updating the difficulty level for this problem from “easy” to “medium” in the tagline. Enjoy your reading!

Setting up the Tricky Algebra Problem

To start, I would like to briefly cover an interesting fundamental property of exponents that will be useful for us in solving this puzzle. It is as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: a^b^c = a^(b*c) = a^(c*b) = a^c^b
Exponent rule (bottom-up) — Math illustrated by the author

This property follows from the fact that any number raised to an arbitrary exponent is just the number multiplied by itself the same number of times (as the exponent). For instance, 4² = 4*4.

Similarly, a number raised to the power of a power is nothing but the number raised to the power multiplied by its power. Since multiplication is commutative, we can switch the powers.

If this explanation is too abstract and wordy for you, just check out the illustration below, and you will get it straight away:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information:4^3^2 = (4*4*4)² = (4*4*4)*(4*4*4) = (4*4)*(4*4)*(4*4) = 4^2^3
Example for exponent rule — Math illustrated by the author

Now that we have established the exponent rule, we just need to find a way to use it in our problem. To achieve this, we can just raise both sides of the given equation to the power of 4 as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: (x^x^4)⁴ = 64⁴
Raising both sides to the power of 4 — Math illustrated by the author

This may look appear like an unnecessarily complicated step now. But as we go along this path, it will become clear to you why we choose to proceed in this manner. Now, by applying the exponent rule we just saw, we can transform the equation as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: (x^x⁴)⁴ = 64⁴; Since a^b^c= a^c^b → (x⁴)^(x⁴) = 64⁴
Further simplification — Math illustrated by the author

We now have an equation form that we can work with further. Let us see how.

The Solution to the Tricky Algebra Problem

On the left-hand side of the above equation, we have a number (x⁴) raised to the power of itself. The opportunistic question to ask, then, is whether we can get the right-hand side of the equation in the same form as well. Why don’t we deploy some algebraic manipulation to see where it gets us?

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: (x⁴)^(x⁴) = 64⁴ = (8²)⁴ = 8^(2*4) = 8⁸; Therefore: (x⁴)^(x⁴) = 8⁸
Further algebraic manipulation — Math illustrated by the author

There we go! We now have a number (8) raised to its own power on the right-hand side as well. All of a sudden, we have simplified the equation to the following form:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: (x⁴)^(x⁴) = 8⁸; Therefore, we could say that x⁴ = 8
Arriving at a simpler equation — Math illustrated by the author

By taking the fourth root on both sides, we arrive at the solution to this puzzle as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: The solution is x = plus or minus the fourth root of eight.
The solution (from incorrect procedure) — Math illustrated by the author

Corrected Solution — Initial Approach

We start by substituting x⁴ with the variable ‘u’. Following this, if we take the logarithm of 2 on both sides of the resulting equation, we would arrive at the following result:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: x^x^4=64; Let x⁴ = u → Therefore: x = u^(1/4); →[u^(1/4)]^u = 64; u^(u/4)=64; Taking logarithm of base 2 on both sides: log_base2_of[u^(u/4)]=log_base2_of(64); →(u/4)*log_base2_of(u) = [6*log_base2_of (2)] [Because log_base2_of(a^b) = b*log_base2_of(a); 2⁶=64]; →(u/4)*log_base2_of(u)=6
Corrected solution (top-down procedure) — Math illustrated by the author

We could further simplify this equation by first multiplying it by 4 on both sides, and then, raising each side to the power of 2 (to eliminate the logarithm we introduced earlier). This procedure would play out as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: (u/4)*log_base2_of(u)=6; Multiplying by 4 on both sides: u*log_base2_of(u)=24; →log_base2_of(u^u)=24 [because log_base2_of(a^b) = b*log_base2_of(a)]; Raising 2 to the power of each side: 2^(log_base2_of(u^u)) = 2²⁴; → u^u = 2²⁴ [Because 2^(log_base2_of(a^b)) = a^b]
Corrected solution (further simplification) — Math illustrated by the author

Corrected Solution to the Tricky Algebra Problem — Closure

We now have an equation form that would directly lead us to the value of ‘u’. All we have to do is algebraically manipulate the right-hand side of the equation such that it is a number raised to the power of itself. It turns out that this number is 8:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: u^u = 2²⁴ = 2^(3*8) = 8⁸ (because 2³ = 8); Therefore: u = 8;
Solution for the variable ‘u’ — Math illustrated by the author

Before we celebrate, we still need to substitute the u = x⁴ to arrive at the final solution in terms of ‘x’. When we proceed with the substitution, we quickly arrive at the final (corrected) solution as follows:

How To Really Solve This Tricky Algebra Problem? (X) — whiteboard graphics showing the following information: u = 8; Substituting x = x⁴, we get: x⁴ = 8. Therefore, the solution is x = plus or minus the fourth root of eight.
Arriving at a simpler equationThe final (corrected) solution — Math illustrated by the author

So, in the end, both approaches lead to the solution of the fourth root of 8. But the corrected solution follows the conventional procedure for solving towers (equations of the kind presented in this essay).

It is worth noting that both approaches cannot lead to the same solution. For instance, try solving 2^3^4 using bottom-up and top-down approaches, and you will get different results.

My take is that I made a mistake with the bottom-up approach. One cannot equate [x^(4)]^[x^(4)] to 8⁸. Under the exponent rule I illustrated, the left-hand side is equivalent to x^(16x). And this expression does not lead to an ‘x⁴’ value of ‘8’.

Final Thoughts

In the end, this puzzle revealed how tricky solving exponential towers can get. We managed to solve the puzzle top-down by employing substitutions and logarithms.

Furthermore, in the context of such exponential problems, it is worth noting the following important point:

When the exponential tower is written explicitly (as shown in the title image), the convention is to solve the expression top-down. However, when the exponential expression is written using carets (for example: a^b^c), there is no convention.

In such a case, both top-down and bottom-up work, and the mathematical world is divided. I have captured this edge case in my essay on order of operations. If you are interested in learning more about it, check it out.

Finally, I hope you enjoyed solving this puzzle as much as I did. I will be continuing to write about interesting and tricky algebra puzzles in the future. So, if that sort of thing interests you, keep an eye on this space in the future!

Acknowledgement: I’d like to thank Hein de Haan for helping with identifying the errors and peer-review.

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